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By El-Kébir Boukas, Roland P. Malhamé

Research, keep watch over and Optimization of complicated Dynamic structures gathers in one quantity a spectrum of advanced dynamic platforms comparable papers written through specialists of their fields, and strongly consultant of present examine traits. complicated platforms current very important demanding situations, in nice half because of their sheer dimension which makes it tricky to know their dynamic habit, optimize their operations, or research their reliability. but, we are living in an international the place, because of expanding inter-dependencies and networking of structures, complexity has develop into the norm. With this in brain, the amount contains elements. the 1st half is devoted to a spectrum of advanced difficulties of selection and keep watch over encountered within the zone of creation and stock platforms. the second one half is devoted to massive scale or multi-agent approach difficulties happening in different components of engineering comparable to telecommunication and electrical energy networks, in addition to extra typical context.

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In this paper it will be assumed that the managing top level decision-making has determined that the best policy to follow is one where the stock level is kept as close as possible to zero. Therefore, defining the 132-norm of a 132 integrable signal z to be the production control problem can now be stated as follows: Switched production control problem. 6)) where d" is a constant known factor, design the production rate (control input) u(t) such that the stock level xl(t) converges to zero when there is no time-varying demand component w(t) and such that it is kept as close as possible to zero when w(t) # 0.

The state space of a ( . ) is M = { I , . . , l } . Let 5 be a random variable uniformly distributed on [O, 11 that is independent of {a;). For each j = 1,.. )and a " ( . ) as aE(t)= a; and a" ( t )= a;, for t E [ne, n~ + E). Note that the state space of a E ( t )(resp, a;) is M = ( 1 , . . ,1). In addition, P ( a ; = k I a; = s,j) = a;. As shown in Yin et al. ) converges weakly to a ( . ) generated by Q,. 9) where [TIE]denotes the integer part of TIE. )(i) is the ith component of the vector ~ ((I),f .

6) for each k. Nevertheless, we still need the aperiodicity to 48 ANALYSIS, CONTROL, AND OPTIMIZATION guarantee the asymptotic expansion of probability vectors; see Yin and Zhang (2000) for details. Moreover, since all the eigenvalues of P* are inside the unit circle, ( P * - I ) is invertible. Define ak = ( aICl , ... ,am*)' = -(P* - I ) - ' P Z ~ , for k = 1 . 7) > where I, = ( 1 , . . , I ) ' E R k X 1 . As shown in Liu et al. (2001), a! 0, and EL=, a; = 1 for each j = 1 , . . ,m,.

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